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## Main Question or Discussion Point

Hello :)

I am an engineer and I am trying to analyse a system which basically contains a cylindrical body free-falling through a body of static water beginning with zero velocity. I am ultimately trying to find what the velocity of the object would be at a depth of 20m. In order to do this I have a spreadsheet set-up which calculates the time derivative velocity, displacement and acceleration terms at any given time.

The basic equation from which I have derived the time dependant derivatives is as follows:

Fg-Fd-Fb = m.a (1)

Where:

Fg = Bodies 'dry' weight

Fd = Drag load

Fb = Buoyancy Force

The equation can be arrange such that:

terminal velocity, v(term) = (m/b)*g (2)

where b is the damping factor.

It is this damping factor which I am uncertain of. According to Stoke's experiments with a small sphere

b = 6*pi*mu*r (3)

where mu =dynamic viscosity

However this equation does not apply to a cylindrical body is not producing sensible answers.

I have considered using the equation for drag forces to calculate this as folloiws:

Fd=0.5*rho*A*Cd*v^2 (4)

so that, damping factor b becomes:

b = 0.5*rho*A*Cd*v (5)

but by using this, the damping factor becomes velocity dependent and therefore varies with time. Therefore the terminal velocity (in accordance with eq. (2)) varies with time which doesn't make sense!

Has anybody done any similar analysis to this or know what the best approach is to take when solving such a problem? Any help would be greatly appreciated as this has had be stumped for a couple of days now!

Thanks in advance,

Ian :)

I am an engineer and I am trying to analyse a system which basically contains a cylindrical body free-falling through a body of static water beginning with zero velocity. I am ultimately trying to find what the velocity of the object would be at a depth of 20m. In order to do this I have a spreadsheet set-up which calculates the time derivative velocity, displacement and acceleration terms at any given time.

The basic equation from which I have derived the time dependant derivatives is as follows:

Fg-Fd-Fb = m.a (1)

Where:

Fg = Bodies 'dry' weight

Fd = Drag load

Fb = Buoyancy Force

The equation can be arrange such that:

terminal velocity, v(term) = (m/b)*g (2)

where b is the damping factor.

It is this damping factor which I am uncertain of. According to Stoke's experiments with a small sphere

b = 6*pi*mu*r (3)

where mu =dynamic viscosity

However this equation does not apply to a cylindrical body is not producing sensible answers.

I have considered using the equation for drag forces to calculate this as folloiws:

Fd=0.5*rho*A*Cd*v^2 (4)

so that, damping factor b becomes:

b = 0.5*rho*A*Cd*v (5)

but by using this, the damping factor becomes velocity dependent and therefore varies with time. Therefore the terminal velocity (in accordance with eq. (2)) varies with time which doesn't make sense!

Has anybody done any similar analysis to this or know what the best approach is to take when solving such a problem? Any help would be greatly appreciated as this has had be stumped for a couple of days now!

Thanks in advance,

Ian :)